I do coding work for a US Fortune 500 company in the financial service sector. On a lark, a coworker asked if I could personally write them a program to automate some audio processing for their job (this has absolutely nothing to do with my job responsibilities). In my spare time, I wrote a quick and dirty solution that worked, a total of 150 lines of Python using open source tools. We asked if it could be sponsored as an internal project and were rejected because audio processing has nothing to do with the company's products, goals, or any market we're even tangentially related to.
I am considering starting from scratch and developing the project in my spare time, and opening a business to sell it. To reiterate, I know I can't use any of the code I've already written.
I talked to my department director about this and he said it'd be a good learning experience to figure out how to set up billing, do my own marketing, customer service, etc. However, I don't have this in writing, and frankly don't care if I did, because if this is something the company would come after me for I don't want any part of it. I love my job and this would only be a small source of side income.
My question: Is it unethical to spin out this project into a private LLC, given the original idea came from a coworker and I've already shared working code with them? Is this acceptable behavior in the world of business?